© Wuhan University 2025

0 Introduction

The study of the stability of constrained mechanical systems plays a significant direction of analytical mechanics^[1-2]. A gradient system is a mathematical system composed of a special class of differential equations. By means of Lyapunov function, this system enables the analysis of both the integrability of the equations and the stability of their solutions^[3-4]. Gradient systems possess a number of important properties. When the equations of mechanical systems are expressible as gradient systems, we use the properties of gradient systems to analyze mechanical systems. Mei and Wu^[5-6] classified gradient systems into four types of basic gradient systems that do not depend on time and two types of generalized gradient systems that explicitly contain time $t$. If the potential function or coefficient matrix of a gradient system explicitly contains $t$, it is called the generalized gradient system^[7-11], which is the extension of gradient systems^[12-15]. Chen et al^[16] systematized nonholonomic systems into generalized gradient systems, and studied the stability of zero-solution of non-autonomous nonholonomic systems. So far, some results have been obtained in the study of mechanical system stability and generalized gradient systems. However, the study of gradient method for non-autonomous Herglotz-type equations has not been reported.

Herglotz-type equations are a class of equations derived from the Herglotz variational principle for non-conservative systems^[17-21]. Donchev^[22] and Lazo et al^[23-24] applied Herglotz-type equations to some well-known non-conservative equations, such as electron beam propagation equation, nonlinear Schrödinger equation, non-conservative Schrödinger equation, vibration string under viscous action, etc. Georgieva et al^[25] studied Noether's theorems of Herglotz type. Recently, considerable progress has been given in the study of Herglotz-type equations for constrained mechanical systems and their Noether theorems^[26-33]. Recently, Zhang^[34] studied constrained Herglotz equations for autonomous nonconservative systems subject to nonholonomic constraints, which are transformed into gradient systems, and then studied the stability of their solutions. In this paper, the generalized gradient method for stability analysis of non-autonomous non-conservative mechanical systems is proposed.

Here is the layout of this paper: In Section 1, the Herglotz-type Lagrange equations and Herglotz-type Hamilton canonical equations for non-autonomous non-conservative systems are listed. In Section 2, a brief overview of two classes of generalized gradient systems (seven forms) is provided. In Section 3, the conditions for transforming non-conservative Herglotz equations into generalized gradient systems are presented. In Section 4, the solutions of Herglotz-type equations and their stability are analyzed. In Section 5, eight specific examples are provided to detail the application of the research findings. Section 6 is the conclusion.

1 Herglotz-Type Equations

For a holonomic nonconservative system, the Herglotz-type Lagrange equations are^[21]

$\frac{\partial L}{\partial q_s}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial {\dot{q}}_s}+\frac{\partial L}{\partial z}\frac{\partial L}{\partial {\dot{q}}_s}=\mathrm{0},s=\mathrm{1,2},\cdots ,n$(1)

where $L=L\left(t,\mathit{q},\dot{\mathit{q}},z\right)$ is the Herglotz-type Lagrangian, $\mathit{q}=\left(q_{\mathrm{1}},q_{\mathrm{2}},\cdots ,q_n\right)$ are the generalized coordinates, $z$ is the Herglotz action and satisfies

$\dot{z}=L\left(t,\mathit{q},\dot{\mathit{q}},z\right)$(2)

Assuming the system is non-singular, we have

$\mathrm{d}\mathrm{e}\mathrm{t}\left(\frac{{\partial }^{\mathrm{2}}L}{\partial {\dot{q}}_s\partial {\dot{q}}_k}\right)\ne \mathrm{0},s,k=\mathrm{1,2},\cdots ,n.$(3)

From equation (1), the generalized accelerations can be given as

${\ddot{q}}_s={\alpha }_s\left(t,\mathit{q},\dot{\mathit{q}},z\right),\hspace{1em}s=\mathrm{1,2},\cdots ,n.$(4)

Let

$a^s=q_s, a^{n+s}={\dot{q}}_s,\hspace{1em}s=\mathrm{1,2},\cdots ,n$(5)

thus equation (4) can be written as

${\dot{a}}^{\mu }=F_{\mu }\left(t,\mathit{a},z\right),\mu =\mathrm{1,2},\cdots ,\mathrm{2}n,$(6)

in which

$F_s=a^{n+s}, F_{n+s}={\alpha }_s, \mathit{a}=\left(a^{\mathrm{1}},a^{\mathrm{2}},\cdots ,a^{\mathrm{2}n}\right)$(7)

By introducing generalized momentum and the Hamiltonian

$p_s=\frac{\partial L}{\partial {\dot{q}}_s},\hspace{1em}H\left(t,\mathit{q},\mathit{p},z\right)=p_s{\dot{q}}_s-L\left(t,\mathit{q},\dot{\mathit{q}},z\right)$(8)

where $\mathit{p}=\left(p_{\mathrm{1}},p_{\mathrm{2}},\cdots ,p_n\right)$. Then equation (1) can be reduced to the Herglotz-type Hamilton canonical equations, i.e,

${\dot{q}}_s=\frac{\partial H}{\partial p_s},\hspace{1em}{\dot{p}}_s=-\frac{\partial H}{\partial q_s}-p_s\frac{\partial H}{\partial z},s=\mathrm{1,2},\cdots ,n.$(9)

We can rewrite equation (9) as

${\dot{a}}^{\mu }={\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu },\hspace{1em}\mu ,\nu =\mathrm{1,2},\cdots ,\mathrm{2}n$(10)

where

$a^s=q_s,\hspace{1em}{a}^{n+s}=p_s,$

$\left({\omega }^{\mu \nu }\right)=\left(\begin{array}{cc}{\mathrm{0}}_{n\times n}& {\mathrm{1}}_{n\times n}\\ -{\mathrm{1}}_{n\times n}& {\mathrm{0}}_{n\times n}\end{array}\right),$

${\mathrm{\Lambda }}_s=\mathrm{0},\hspace{1em}{\mathrm{\Lambda }}_{n+s}=-a^{n+s}\frac{\partial H}{\partial z}$(11)

2 Two Classes of Generalized Gradient Systems

We introduce two classes of generalized gradient systems in detail. The first is the gradient system whose the $V$ function contains $t$. The second is the gradient system in which both the $V$ function and the coefficient matrix (i.e., $\left({\zeta }_{sk}\right)$, $\left({\varpi }_{sk}\right)$ and $\left({\sigma }_{sk}\right)$) contain $t$.

2.1 Generalized Gradient System I

Let the potential function be $V\left(t,\mathit{x}\right)$ and have

${\dot{x}}_s=-\frac{\partial V\left(t,\mathit{x}\right)}{\partial x_s}$(12)

then equation (12) are called the generalized gradient system I-1, where $s=\mathrm{1,2},\cdots ,n$ and $\mathit{x}=\left(x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_n\right)$.

Let the potential function be $V\left(t,\mathit{x}\right)$ and have

${\dot{x}}_s={\zeta }_{sk}\left(\mathit{x}\right)\frac{\partial V\left(t,\mathit{x}\right)}{\partial x_k}$(13)

thus equation (13) are called the generalized gradient system I-2, where $\left({\zeta }_{sk}\left(\mathit{x}\right)\right)$ is an antisymmetric matrix with ${\zeta }_{sk}\left(\mathit{x}\right)=-{\zeta }_{ks}\left(\mathit{x}\right)$.

Let the potential function be $V\left(t,\mathit{x}\right)$ and have

${\dot{x}}_s={\varpi }_{sk}\left(\mathit{x}\right)\frac{\partial V\left(t,\mathit{x}\right)}{\partial x_k}$(14)

then equation (14) are called the generalized gradient system I-3, where $\left({\varpi }_{sk}\left(\mathit{x}\right)\right)$ is a symmetric negative definite matrix.

Let the potential function be $V\left(t,\mathit{x}\right)$ and have

${\dot{x}}_s={\sigma }_{sk}\left(\mathit{x}\right)\frac{\partial V\left(t,\mathit{x}\right)}{\partial x_k}$(15)

then equation (15) are called the generalized gradient system Ⅰ-4, where $\left({\sigma }_{sk}\left(\mathit{x}\right)\right)$ is a semi-negative definite matrix.

2.2 Generalized Gradient System II

Let the potential function be $V\left(t,\mathit{x}\right)$ and have

${\dot{x}}_s={\zeta }_{sk}\left(t,\mathit{x}\right)\frac{\partial V\left(t,\mathit{x}\right)}{\partial x_k}$(16)

thus equation (16) are called the generalized gradient system Ⅱ-1, where $\left({\zeta }_{sk}\left(t,\mathit{x}\right)\right)$ is an antisymmetric matrix with ${\zeta }_{sk}\left(t,\mathit{x}\right)=-{\zeta }_{ks}\left(t,\mathit{x}\right)$, and $s,k=\mathrm{1,2},\cdots ,n$.

Let the potential function be $V\left(t,\mathit{x}\right)$ and have

${\dot{x}}_s={\varpi }_{sk}\left(t,\mathit{x}\right)\frac{\partial V\left(t,\mathit{x}\right)}{\partial x_k}$(17)

then equation (17) are called the generalized gradient system Ⅱ-2, where $\left({\varpi }_{sk}\left(t,\mathit{x}\right)\right)$ is a symmetric negative definite matrix.

Let the potential function be $V\left(t,\mathit{x}\right)$ and have

${\dot{x}}_s={\sigma }_{sk}\left(t,\mathit{x}\right)\frac{\partial V\left(t,\mathit{x}\right)}{\partial x_k}$(18)

thus equation (18) are called the generalized gradient system Ⅱ-3, where $\left({\sigma }_{sk}\left(t,\mathit{x}\right)\right)$ is a semi-negative definite matrix.

3 Generalized Gradient Representation of Herglotz-Type Equations

Herglotz-type equations (6) and (10) are generally not generalized gradient systems. The conditions for Herglotz-type equations to be transformed into the above generalized gradient systems are as follows.

In this section, let $\mu ,\nu ,\rho =\mathrm{1,2},\cdots ,\mathrm{2}n$.

3.1 Herglotz-Type Equations and Generalized Gradient System I

For equation (6), if the conditions

$\frac{\partial F_{\mu }}{\partial a^{\nu }}-\frac{\partial F_{\nu }}{\partial a^{\mu }}=\mathrm{0}$(19)

are satisfied, they can be reduced to the generalized gradient system I-1, whose $V$ function is determined by

$F_{\mu }=-\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\mu }}$(20)

Similarly, for equation (10), if the conditions

$\frac{\partial }{\partial a^{\rho }}\left({\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu }\right)-\frac{\partial }{\partial a^{\mu }}\left({\omega }^{\rho \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\rho }\right)=\mathrm{0}$(21)

are satisfied, they can be reduced to the generalized gradient system I-1, whose $V$ function is determined by

${\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu }=-\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\mu }}$(22)

For equation (6), if there is a matrix $\left({\zeta }_{\mu \nu }\left(\mathit{a}\right)\right)$ and a function $V$ that satisfy

$F_{\mu }={\zeta }_{\mu \nu }\left(\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(23)

they can be reduced to the generalized gradient system I-2.

Similarly, for equation (10), if there is a matrix $\left({\zeta }_{\mu \nu }\left(\mathit{a}\right)\right)$ and a function $V$ that satisfy

${\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu }={\zeta }_{\mu \nu }\left(\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(24)

they can be reduced to the generalized gradient system I-2.

For equation (6), if there is a matrix $\left({\varpi }_{\mu \nu }\left(\mathit{a}\right)\right)$ and a function $V$ that satisfy

$F_{\mu }={\varpi }_{\mu \nu }\left(\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(25)

they can be reduced to the generalized gradient system I-3.

Similarly, for equation (10), if there is a matrix $\left({\varpi }_{\mu \nu }\left(\mathit{a}\right)\right)$ and a function $V$ that satisfy

${\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu }={\varpi }_{\mu \nu }\left(\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(26)

they can be reduced to the generalized gradient system I-3.

For equation (6), if there is a matrix $\left({\sigma }_{\mu \nu }\left(\mathit{a}\right)\right)$ and a function $V$ that satisfy

$F_{\mu }={\sigma }_{\mu \nu }\left(\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }},$(27)

they can be reduced to the generalized gradient system I-4.

Similarly, for equation (10), if there is a matrix $\left({\sigma }_{\mu \nu }\left(\mathit{a}\right)\right)$ and a function $V$ that satisfy

${\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu }={\sigma }_{\mu \nu }\left(\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(28)

they can be reduced to the generalized gradient system I-4.

3.2 Herglotz-Type Equations and Generalized Gradient System II

For equation (6), if there exists a matrix $\left({\zeta }_{\mu \nu }\left(t,\mathit{a}\right)\right)$ and a function $V$ that satisfy

$F_{\mu }={\zeta }_{\mu \nu }\left(t,\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(29)

they can be reduced to the generalized gradient system Ⅱ-1.

Similarly, for equation (10), if there exists a matrix $\left({\zeta }_{\mu \nu }\left(t,\mathit{a}\right)\right)$ and a function $V$ that satisfy

${\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu }={\zeta }_{\mu \nu }\left(t,\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(30)

they can be reduced to the generalized gradient system Ⅱ-1.

For equation (6), if there exists a matrix $\left({\varpi }_{\mu \nu }\left(t,\mathit{a}\right)\right)$ and a function $V$ that satisfy

$F_{\mu }={\varpi }_{\mu \nu }\left(t,\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(31)

they can be reduced to the generalized gradient system Ⅱ-2.

Similarly, for equation (10), if there exists a matrix $\left({\varpi }_{\mu \nu }\left(t,\mathit{a}\right)\right)$ and a function $V$ that satisfy

${\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu }={\varpi }_{\mu \nu }\left(t,\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(32)

they can be reduced to the generalized gradient system Ⅱ-2.

For equation (6), if there exists a matrix $\left({\sigma }_{\mu \nu }\left(t,\mathit{a}\right)\right)$ and a function $V$ that satisfy

$F_{\mu }={\sigma }_{\mu \nu }\left(t,\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(33)

they can be reduced to the generalized gradient system Ⅱ-3.

Similarly, for equation (10), if there exists a matrix $\left({\sigma }_{\mu \nu }\left(t,\mathit{a}\right)\right)$ and a function $V$ that satisfy

${\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}+{\mathrm{\Lambda }}_{\mu }={\sigma }_{\mu \nu }\left(t,\mathit{a}\right)\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\nu }}$(34)

they can be reduced to the generalized gradient system Ⅱ-3.

However, it should be noted that conditions (19), (21), (23), (24), (25), (26), (27), (28), (29), (30), (31), (32), (33) and (34) are sufficient. If the conditions are not satisfied, it is still not possible to conclude that the system is not a generalized gradient system, because this is related to the choice of the first-order form obtained by Herglotz equations.

4 Stability of Zero Solution of Herglotz-Type Equations

In this section, we denote $V^{+}\left(t,\mathit{x}\right)$ as positive definite, $V^{-}\left(t,\mathit{x}\right)$ as negative definite, and $V^{\mathrm{0}-}\left(t,\mathit{x}\right)$ as semi-negative definite.

The theories for studying the stability of the zero solution are as follows: First, if $V^{+}$ exists as well as ${\dot{V}}^{\mathrm{0}-}$, then the zero solution of the system is stable. Second, if $V^{+}$ exists as well as ${\dot{V}}^{-}$, then the zero solution of the system is asymptotically stable. Third, if $V^{+}$ exists as well as ${\dot{V}}^{\mathrm{0}-}$, and the function $V$ has an infinitesimal upper bound, then the zero solution of the system is uniformly stable. Fourth, if $V^{+}$ exists as well as ${\dot{V}}^{-}$, and the function $V$ has an infinitesimal upper bound, then the zero solution of the system is uniformly asymptotically stable.

4.1 Generalized Gradient System I

According to equation (12), we get the derivative of $V$, i.e.,

$\dot{V}=\frac{\partial V}{\partial t}-\frac{\partial V}{\partial x_s}\frac{\partial V}{\partial x_s}$(35)

In equation (35), there is $-\frac{\partial V}{\partial x_s}\frac{\partial V}{\partial x_s}<\mathrm{0}$. If $V^{+}$ exists as well as ${\dot{V}}^{-}$, thus the solution of the system is asymptotically stable.

According to equation (13), we obtain the derivative of $V$, i.e.,

$\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_s}{\zeta }_{sk}\frac{\partial V}{\partial x_k}=\frac{\partial V}{\partial t}$(36)

It can be shown that if $V^{+}$ exists as well as $\frac{\partial V}{\partial t}\le \mathrm{0}$, then the solution is stable.

According to equation (14), we obtain the derivative of $V$, i.e.,

$\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_s}{\varpi }_{sk}\frac{\partial V}{\partial x_k}$(37)

In equation (37), there is $\frac{\partial V}{\partial x_s}{\varpi }_{sk}\frac{\partial V}{\partial x_k}<\mathrm{0}$. The solution is asymptotically stable when $V^{+}$ exists as well as ${\dot{V}}^{-}$.

According to equation (15), we obtain the derivative of $V$, i.e.,

$\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_s}{\sigma }_{sk}\frac{\partial V}{\partial x_k}.$(38)

In equation (38), there is $\frac{\partial V}{\partial x_s}{\sigma }_{sk}\frac{\partial V}{\partial x_k}\le \mathrm{0}$.

4.2 Generalized Gradient System II

According to equation (16), we obtain the derivative of $V$, i.e.,

$\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_s}{\zeta }_{sk}\left(t,\mathit{x}\right)\frac{\partial V}{\partial x_k}=\frac{\partial V}{\partial t}$(39)

It can be seen that if $V^{+}$ exists as well as $\frac{\partial V}{\partial t}<\mathrm{0}$, then the solution is stable.

According to equation (17), we obtain the derivative of $V$, i.e.,

$\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_s}{\varpi }_{sk}\left(t,\mathit{x}\right)\frac{\partial V}{\partial x_k}$(40)

In equation (40), there is $\frac{\partial V}{\partial x_s}{\varpi }_{sk}\left(t,\mathit{x}\right)\frac{\partial V}{\partial x_k}<\mathrm{0}$.

According to equation (18), we obtain the derivative of $V$, i.e.,

$\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_s}{\sigma }_{sk}\left(t,\mathit{x}\right)\frac{\partial V}{\partial x_k}$(41)

In equation (41), there is $\frac{\partial V}{\partial x_s}{\sigma }_{sk}\left(t,\mathit{x}\right)\frac{\partial V}{\partial x_k}\le \mathrm{0}$.

5 Examples

The conditions for the transformation of Herglotz-type equations (two forms) of non-autonomous non-conservative systems into two classes of generalized gradient systems (seven forms) are given above, and the stability of their zero solutions is discussed. For each case, we give examples to demonstrate in detail.

Example 1 Consider the Herglotz-type Hamiltonian

$H=-\mathrm{2}pq\left(\mathrm{1}+t\right)+\left(\mathrm{4}+\mathrm{2}t\right)z$(42)

where $\frac{\mathrm{d}z}{\mathrm{d}t}=p\dot{q}-H$. Transform equation (42) into generalized gradient system I-1 and analyze its stability.

According to equation (9), the Herglotz-type canonical equations are

$\dot{q}=-\mathrm{2}\left(\mathrm{1}+t\right)q,\hspace{1em}\dot{p}=-\mathrm{2}p$(43)

Let

$a^{\mathrm{1}}=q,\hspace{1em}{a}^{\mathrm{2}}=p,$(44)

then

${\dot{a}}^{\mathrm{1}}=-\mathrm{2}\left(\mathrm{1}+t\right)a^{\mathrm{1}},\hspace{1em}{\dot{a}}^{\mathrm{2}}=-\mathrm{2}{a}^{\mathrm{2}}$(45)

It can be readily verified that it meets the specified condition (21). Therefore equation (45) can be formulated as the generalized gradient system I-1 whose $V$ function is determined by

$-\mathrm{2}\left(\mathrm{1}+t\right)a^{\mathrm{1}}=-\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\mathrm{1}}},\hspace{1em}-a^{\mathrm{2}}=-\frac{\partial V\left(t,\mathit{a}\right)}{\partial a^{\mathrm{2}}}$(46)

Solving equation (46), we get

$V=\left(\mathrm{1}+t\right){\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}.$(47)

Then we have

$\dot{V}=-{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\left[\mathrm{4}{\left(\mathrm{1}+t\right)}^{\mathrm{2}}-\mathrm{1}\right]-\mathrm{4}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}$(48)

In the domain of $a^{\mathrm{1}}=a^{\mathrm{2}}=\mathrm{0}$, $t\ge \mathrm{0}$, $V^{+}$ exists as well as ${\dot{V}}^{-}$, therefore, the zero solution $a^{\mathrm{1}}=a^{\mathrm{2}}=\mathrm{0}$ is asymptotically stable.

Example 2 Consider the Herglotz-type Lagrangian

$L=\mathrm{2}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)q^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}{\dot{q}}^{\mathrm{2}}-\frac{{\mathrm{e}}^{-t}}{\mathrm{1}+{\mathrm{e}}^{-t}}z$(49)

Transform equation (49) into generalized gradient system I-2 and analyze its stability.

The Herglotz-type Lagrange equations give

$\ddot{q}=-\mathrm{4}q\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)-\frac{{\mathrm{e}}^{-t}}{\mathrm{1}+{\mathrm{e}}^{-t}}\dot{q}$(50)

Let

$a^{\mathrm{1}}=q,\hspace{1em}{a}^{\mathrm{2}}=\frac{\dot{q}}{\mathrm{2}+\mathrm{2}{\mathrm{e}}^{-t}}$(51)

equation (50) can be expressed in first-order form

${\dot{a}}^{\mathrm{1}}=\mathrm{2}{a}^{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right),\hspace{1em}{\dot{a}}^{\mathrm{2}}=-\mathrm{2}{a}^{\mathrm{1}}$(52)

It can be formulated as generalized gradient system I-2, i.e.,

$\left(\begin{array}{c}{\dot{a}}^{\mathrm{1}}\\ {\dot{a}}^{\mathrm{2}}\end{array}\right)=\left(\begin{array}{cc}\mathrm{0}& \mathrm{1}\\ -\mathrm{1}& \mathrm{0}\end{array}\right)\left(\begin{array}{c}\frac{\partial V}{\partial a^{\mathrm{1}}}\\ \frac{\partial V}{\partial a^{\mathrm{2}}}\end{array}\right)$(53)

where

$V={\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)$(54)

Using equation (52), we get

$\dot{V}=-{\mathrm{e}}^{-t}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}$(55)

Since $V^{+}$ exists as well as ${\dot{V}}^{\mathrm{0}-}$. Then solution $a^{\mathrm{1}}=a^{\mathrm{2}}=\mathrm{0}$ is stable.

Example 3 Consider the Herglotz-type Hamiltonian

$H=-q^{\mathrm{2}}-\mathrm{2}pq+p^{\mathrm{2}}\frac{\mathrm{1}}{\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t}+\left(\frac{\mathrm{4}}{\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t}+\mathrm{2}\right)z$(56)

where $\frac{\mathrm{d}z}{\mathrm{d}t}=p\dot{q}-H$. Transform this system into generalized gradient system I-3 and analyze its stability.

Herglotz-type canonical equations give

$\dot{q}=-\mathrm{2}q+\frac{\mathrm{2}p}{\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t},\hspace{1em}\dot{p}=\mathrm{2}q-\frac{\mathrm{4}p}{\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t}$(57)

Let

$a^{\mathrm{1}}=q,\hspace{1em}{a}^{\mathrm{2}}=p$(58)

we rewrite equation (57) as

${\dot{a}}^{\mathrm{1}}=-\mathrm{2}{a}^{\mathrm{1}}+\frac{\mathrm{2}{a}^{\mathrm{2}}}{\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t},\hspace{1em}{\dot{a}}^{\mathrm{2}}=\mathrm{2}{a}^{\mathrm{1}}-\frac{\mathrm{4}{a}^{\mathrm{2}}}{\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t}$(59)

which can be reduced to generalized gradient system I-3, i.e.,

$\left(\begin{array}{c}{\dot{a}}^{\mathrm{1}}\\ {\dot{a}}^{\mathrm{2}}\end{array}\right)=\left(\begin{array}{cc}-\mathrm{1}& \mathrm{1}\\ \mathrm{1}& -\mathrm{2}\end{array}\right)\left(\begin{array}{c}\frac{\partial V}{\partial a^{\mathrm{1}}}\\ \frac{\partial V}{\partial a^{\mathrm{2}}}\end{array}\right)$(60)

where

$V={\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\frac{{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}}{\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t}$(61)

Using equation (59), we get

$\dot{V}=-\mathrm{4}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}-\frac{{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}}{{\left(\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t\right)}^{\mathrm{2}}}\left(\mathrm{8}-\mathrm{s}\mathrm{i}\mathrm{n}t\right)+\frac{\mathrm{8}{a}^{\mathrm{1}}{a}^{\mathrm{2}}}{\mathrm{2}+\mathrm{c}\mathrm{o}\mathrm{s}t}$(62)

Since $V^{+}$ exists as well as ${\dot{V}}^{-}$, then solution $a^{\mathrm{1}}=a^{\mathrm{2}}=\mathrm{0}$ is asymptotically stable.

Example 4 Consider the Herglotz-type Lagrangian

$L=\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{e}}^{-t}{q}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right){\dot{q}}^{\mathrm{2}}-\frac{\mathrm{2}+\mathrm{5}{\mathrm{e}}^{-t}+{\mathrm{e}}^{-\mathrm{2}t}}{\mathrm{1}+{\mathrm{e}}^{-t}}z$(63)

Transform equation (63) into generalized gradient system I-4 and analyze its stability.

Herglotz-type equations give

$\ddot{q}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)+\left(\mathrm{1}+\mathrm{2}{\mathrm{e}}^{-t}\right)\dot{q}+{\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)}^{\mathrm{2}}\dot{q}+{\mathrm{e}}^{-t}q=\mathrm{0}$(64)

Let

$a^{\mathrm{1}}=q,\hspace{1em}{a}^{\mathrm{2}}=\frac{\dot{q}+q}{\mathrm{1}+{\mathrm{e}}^{-t}}$(65)

equation (64) can be expressed in first-order form

${\dot{a}}^{\mathrm{1}}=-a^{\mathrm{1}}+a^{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right),\hspace{1em}{\dot{a}}^{\mathrm{2}}=a^{\mathrm{1}}-a^{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)$(66)

Take

$V=\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right),\hspace{1em}\left({\sigma }_{ks}\right)=\left(\begin{array}{cc}-\mathrm{1}& \mathrm{1}\\ \mathrm{1}& -\mathrm{1}\end{array}\right)$(67)

Using equation (66), we get

$\begin{array}{l}\dot{V}=-{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right){\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}\left[\mathrm{1}+\mathrm{2}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)\right]\\ +\mathrm{2}{a}^{\mathrm{1}}{a}^{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)\end{array}$(68)

Since $V^{+}$ exists as well as ${\dot{V}}^{-}$, then solution $a^{\mathrm{1}}=a^{\mathrm{2}}=\mathrm{0}$ is asymptotically stable.

Example 5 The Herglotz-type Lagrangian

$L={\left(\mathrm{1}+t\right)}^{\mathrm{2}}\mathrm{2}{q}^{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)-\frac{\mathrm{1}}{\mathrm{2}}{\dot{q}}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{1}+t}z$(69)

where $\frac{\mathrm{d}z}{\mathrm{d}t}=L$. Transform equation (69) into generalized gradient system II-1 and analyze its stability.

Herglotz-type equations give

$\ddot{q}=\frac{\mathrm{1}}{\mathrm{1}+t}\dot{q}-{\left({\mathrm{e}}^{-t}+\mathrm{1}\right)}^{\mathrm{2}}\mathrm{4}q\left(t+\mathrm{1}\right)$(70)

Let

$a^{\mathrm{1}}=q,\hspace{1em}{a}^{\mathrm{2}}=\frac{\dot{q}}{\mathrm{2}\left(\mathrm{1}+t\right)}$(71)

Then we can rewrite equation (70) as

${\dot{a}}^{\mathrm{1}}=\mathrm{2}{a}^{\mathrm{2}}\left(\mathrm{1}+t\right),\hspace{1em}{\dot{a}}^{\mathrm{2}}=-\mathrm{2}{a}^{\mathrm{1}}\left(\mathrm{1}+t\right)\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)$(72)

Equation (72) can be formulated as generalized gradient system Ⅱ-1, i.e.,

$\left(\begin{array}{c}{\dot{a}}^{\mathrm{1}}\\ {\dot{a}}^{\mathrm{2}}\end{array}\right)=\left(\begin{array}{cc}\mathrm{0}& \mathrm{1}+t\\ -\left(\mathrm{1}+t\right)& \mathrm{0}\end{array}\right)\left(\begin{array}{c}\frac{\partial V}{\partial a^{\mathrm{1}}}\\ \frac{\partial V}{\partial a^{\mathrm{2}}}\end{array}\right)$(73)

where

$V={\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\left(\mathrm{1}+{\mathrm{e}}^{-t}\right)+{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}$(74)

Using equation (72), we have

$\dot{V}=-{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}{\mathrm{e}}^{-t}<\mathrm{0}$(75)

Since $V^{+}$ exists as well as ${\dot{V}}^{\mathrm{0}-}$, then solution $a^{\mathrm{1}}=a^{\mathrm{2}}=\mathrm{0}$ is uniformly stable.

Example 6 For the non-conservative single degree-of-freedom system, Herglotz-type Hamiltonian is

$H=-\mathrm{2}\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)pq+\mathrm{2}\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{4}\right)z$(76)

Transform it into generalized gradient system Ⅱ-2 and analyze its stability.

Herglotz-type canonical equations give

$\dot{q}=-\mathrm{2}q\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right),\hspace{1em}\dot{p}=-\mathrm{4}p\left(t+\mathrm{1}\right)$(77)

Let

$a^{\mathrm{1}}=q,\hspace{1em}{a}^{\mathrm{2}}=p$(78)

Equation (77) can be expressed as

${\dot{a}}^{\mathrm{1}}=-\mathrm{2}{a}^{\mathrm{1}}\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right),\hspace{1em}{\dot{a}}^{\mathrm{2}}=-\mathrm{4}{a}^{\mathrm{2}}\left(t+\mathrm{1}\right)$(79)

It can be formulated as generalized gradient system Ⅱ-2, i.e.,

$\left(\begin{array}{c}{\dot{a}}^{\mathrm{1}}\\ {\dot{a}}^{\mathrm{2}}\end{array}\right)=\left(\begin{array}{cc}-\left(t+\mathrm{1}\right)& \mathrm{0}\\ \mathrm{0}& -\mathrm{2}\left(t+\mathrm{1}\right)\end{array}\right)\left(\begin{array}{c}\frac{\partial V}{\partial a^{\mathrm{1}}}\\ \frac{\partial V}{\partial a^{\mathrm{2}}}\end{array}\right)$(80)

where

$V={\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\left(\mathrm{2}+\mathrm{s}\mathrm{i}\mathrm{n}t\right)+{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}$(81)

Using equation (79), we have

$\dot{V}=-{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\left[\mathrm{4}\left(t+\mathrm{1}\right)\left(\mathrm{2}+\mathrm{s}\mathrm{i}\mathrm{n}t\right)-\mathrm{c}\mathrm{o}\mathrm{s}t\right]-\mathrm{8}\left(t+\mathrm{1}\right){\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}.$(82)

Since $V^{+}$ exists as well as ${\dot{V}}^{-}$, then solution $a^{\mathrm{1}}=a^{\mathrm{2}}=\mathrm{0}$ is uniformly asymptotically stable.

Example 7 For a non-conservative two-degree-of-freedom system, the Herglotz-type Lagrangian is

$\begin{array}{l}L=\frac{\mathrm{1}}{\mathrm{2}}\left({\dot{q}}_{\mathrm{1}}^{\mathrm{2}}+{\dot{q}}_{\mathrm{2}}^{\mathrm{2}}\right)-\frac{\mathrm{1}}{\mathrm{2}}\left(q_{\mathrm{1}}^{\mathrm{2}}+q_{\mathrm{2}}^{\mathrm{2}}\right)\left(\mathrm{1}+t\right)\left[\mathrm{4}\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)-\mathrm{1}\right]\\ -\mathrm{2}\left[\mathrm{1}+\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)\right]z\end{array}$(83)

Transform it into generalized gradient system Ⅱ-2 and analyze its stability.

Herglotz-type equations are

$\begin{array}{l}{\ddot{q}}_{\mathrm{1}}=-q_{\mathrm{1}}\left(t+\mathrm{1}\right)\left[\mathrm{4}\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)-\mathrm{1}\right]\\ -\mathrm{2}{\dot{q}}_{\mathrm{1}}\left[\mathrm{1}+\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)\right],\end{array}$

$\begin{array}{l}{\ddot{q}}_{\mathrm{2}}=-q_{\mathrm{2}}\left(t+\mathrm{1}\right)\left[\mathrm{4}\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)-\mathrm{1}\right]\\ -\mathrm{2}{\dot{q}}_{\mathrm{2}}\left[\mathrm{1}+\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)\right]\end{array}$(84)

Let

$a^{\mathrm{1}}=q_{\mathrm{1}},\hspace{1em}{a}^{\mathrm{2}}=q_{\mathrm{2}},\hspace{1em}{a}^{\mathrm{3}}={\dot{q}}_{\mathrm{1}}+\mathrm{2}{q}_{\mathrm{1}},\hspace{1em}{a}^{\mathrm{4}}={\dot{q}}_{\mathrm{2}}+\mathrm{2}{q}_{\mathrm{2}}$(85)

We rewrite equation (84) as

${\dot{a}}^{\mathrm{1}}=-\mathrm{2}{a}^{\mathrm{1}}+a^{\mathrm{3}},$

${\dot{a}}^{\mathrm{2}}=-\mathrm{2}{a}^{\mathrm{2}}+a^{\mathrm{4}},$

${\dot{a}}^{\mathrm{3}}=-\left(t+\mathrm{1}\right)\left[\mathrm{2}{a}^{\mathrm{3}}\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)-a^{\mathrm{1}}\right],$

${\dot{a}}^{\mathrm{4}}=-\left(t+\mathrm{1}\right)\left[\mathrm{2}{a}^{\mathrm{4}}\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)-a^{\mathrm{2}}\right]$(86)

It can be formulated as generalized gradient system Ⅱ-2, i.e.,

$\left(\begin{array}{c}{\dot{a}}^{\mathrm{1}}\\ {\dot{a}}^{\mathrm{2}}\\ {\dot{a}}^{\mathrm{3}}\\ {\dot{a}}^{\mathrm{4}}\end{array}\right)=\left(\begin{array}{cccc}-\mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathrm{1}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -\left(t+\mathrm{1}\right)& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& -\left(t+\mathrm{1}\right)\end{array}\right)\left(\begin{array}{c}\frac{\partial V}{\partial a^{\mathrm{1}}}\\ \frac{\partial V}{\partial a^{\mathrm{2}}}\\ \frac{\partial V}{\partial a^{\mathrm{3}}}\\ \frac{\partial V}{\partial a^{\mathrm{4}}}\end{array}\right)$(87)

where

$\begin{array}{l}V={\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}+{\left(a^{\mathrm{3}}\right)}^{\mathrm{2}}\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)\\ +{\left(a^{\mathrm{4}}\right)}^{\mathrm{2}}\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)-a^{\mathrm{1}}{a}^{\mathrm{3}}-a^{\mathrm{2}}{a}^{\mathrm{4}},\end{array}$(88)

Using equation (86), we get

$\begin{array}{l}\dot{V}=-\left(\mathrm{5}+t\right){\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}-\left[\mathrm{1}-\mathrm{c}\mathrm{o}\mathrm{s}t+\mathrm{4}\left(t+\mathrm{1}\right){\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)}^{\mathrm{2}}\right]{\left(a^{\mathrm{3}}\right)}^{\mathrm{2}}\\ \hspace{1em}+\left[\mathrm{4}+\mathrm{4}\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)\right]a^{\mathrm{1}}{a}^{\mathrm{3}}-\left(\mathrm{5}+t\right){\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}\\ \hspace{1em}-\left[\mathrm{1}-\mathrm{c}\mathrm{o}\mathrm{s}t+\mathrm{4}\left(t+\mathrm{1}\right){\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)}^{\mathrm{2}}\right]{\left(a^{\mathrm{4}}\right)}^{\mathrm{2}}\\ \hspace{1em}+\left[\mathrm{4}+\mathrm{4}\left(t+\mathrm{1}\right)\left(\mathrm{s}\mathrm{i}\mathrm{n}t+\mathrm{2}\right)\right]a^{\mathrm{2}}{a}^{\mathrm{4}}\end{array}$(89)